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Morse Theory
The standard procedure:
- Show that \(D\) is a Fredholm operator
- Show that \(D\) is surjective, so \({\mathsf{Ind}}D = \dim \ker D\)
- Show a moduli space is the intersection of some section \(s\) of a bundle with the zero section.
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Show that this intersection is transverse, i.e. \(Ds\) is surjective.
- Vary the Riemannian metric and use a second category theorem to get the Morse-Smale condition.
- Apply the infinite-dimensional inverse function theorem
- Show that a Frechet manifold is in fact a Banach manifold and apply a version of Sard’s Theorem
Motivations
- Can be used to prove the high dimensional case of the generalized Unsorted/Poincare conjectures
Results
Theorem: Every compact smooth manifold admits a Morse function.
Theorem: Morse function are generic: given any smooth function \(f: X\to Y\), there is an arbitrarily small perturbation of \(f\) that is Morse.
See Morse lemma
Theorem 3: If \((W; M_0, M_1) \to I\) is Morse with no critical points then \(W \cong_{\operatorname{Diff}} I \times M_0\)
Theorem: If \(X\) is closed and admits a Morse function with exactly 2 critical points, \(X\) is homeomorphic to \(S^n\).
Possibly used in Milnor’s Unsorted/parahoric 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function)
Theorem: \(M\) is homotopy equivalent to a CW complex with one cell of dimension \(k\) for each critical point of \(f\) of index of a Morse function \(k\).
Gradients
# Energy
Critical points
Idea: the number of linearly independent direction you can move for which the function decreases.
Morse chain complex
Morse inequalities
Broken trajectories
Moduli space of flow lines
Zero set of a section
See vertical and horizontal subspace
Sard
Pictures
Examples
Notes
Dave’s Videos
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Historic note: Morse wanted to know not the diffeomorphism type of \(M\), but rather the homotopy type.
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Definition: critical values and critical points
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Definition: critical point
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Theorem (Smale, h-cobordism theorem
- If \(X^n\) is a smooth cobordism.
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Corollary (High-Dimensional Unsorted/Poincare conjectures
- If \(X_1^n, X_2^n \cong_{\operatorname{Diff}} S^n\), then there exists an h-cobordism between them.
- Proof: use algebraic topology to eliminate (cancel) critical points.
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Definition: index of a Morse function
- Look at coordinate-free def?
- Standard form at critical points
- Alternatively: Hessian is non-singular at every critical point.
- \(f^{-1}{{\partial}}Y) = {{\partial}}X\)
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Definition: Stable and generic
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Definition: cobordism
- Example: (pair of pants)
- Category: Objects are manifolds, morphisms are cobordisms between them
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Consequence of theorem 3: \(M_0 \cong_{\text{Diff}} M_1\) is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare.
- Recall that this is proved by constructing a vector field \(V\) on \(W\), then using a diffeomorphism \(\phi:I \times M_0 \to W\) by flowing along \(V\).
- Can we do gradient flow in the presence of a metric? #todo/questions
Intro Video
https://www.youtube.com/watch?v=78OMJ8JKDqI
Morse theory: handles nice singularities. Can have worse ones, covered by [dynamical systems](catastrophe theory](dynamical%20systems](catastrophe%20theory) (dynamical systems).
Importance of CW complexes: triangulation of surfaces.
See Morse lemma
Morse Theorem 1: If there are no critical points, \(M_A \simeq M_B\).
Stable vs unstable manifolds:
Consider height function on torus. Circles are index 0 critical points, triangle is index 1.
Cancellation:
Can use persistent homology to measure “importance” of critical points.
Unsorted
https://youtu.be/mIUi1zIUQJw?t=42
- Diffeomorphism type depends on isotopy classes of attaching maps.
See handle decomposition
More Notes
Historic note: Morse wanted to know not the diffeomorphism type of \(M\), but rather the homotopy type. - Theorem (Smale, h-cobordism) - If \(X^n\) is a smooth cobordism, \(n\geq 6\), \(\pi_1(X) = 0\), and \(X\) “looks like” a product in algebraic topology, then \(X\) is a product cobordism. - Corollary (High-Dim Poincare) - If \(X_1^n, X_2^n \cong_{\operatorname{Diff}} S^n\), then there exists an \(h{\hbox{-}}\)cobordism between them. - Proof: use algebraic topology to eliminate (cancel) critical points. - Theorem: Every compact manifold has a Morse function. - Theorem: Morse functions are generic (given any smooth function \(f: X\to Y\), there’s an arbitrarily small perturbation of \(f\) that is Morse). - Theorem (Morse Lemma): If \(p\in {\mathbf{R}}^n\) is a critical point of \(f: {\mathbf{R}}^n \to {\mathbf{R}}\) such that the Hessian \(H_f(p)\) is a non-degenerate bilinear form, then \(f\) is locally Morse (standard form). - Theorem: If \((W; M_0, M_1) \to I\) is Morse with no critical points then \(W \cong_{\operatorname{Diff}} I \times M_0\) - Consequence: \(M_0 \cong_{\text{Diff}} M_1\) is a diffeomorphism, useful to show two things are diffeomorphic, used in higher-dimensional Poincare. - Theorem: If \(X\) is closed and admits a Morse function with exactly 2 critical points, \(X\) is homeomorphic to \(S^n\). - Possibly used in Milnor’s exotic 7-sphere (show a diffeomorphism invariant differs but admits such a Morse function) - Diffeomorphism type depends on isotopy classes of attaching maps.
Morse Theory
Goal: handlebody decomposition, or for the purposes of the above theorems, retractions onto a CW complex. Decomposing a cobordism into a sequence of elementary cobordisms (admit a Morse function with a single critical point).
Fact: since \(\phi\) is Morse, \(M^{2n}\) can be retracted onto a complex of dimension \(d\leq n\), since all critical points will have index \(\leq n\).
Note: this immediately implies the Lefschetz Hyperplane theorem for affine manifolds \(N\), i.e. that they are entirely determined by the homology and homotopy of \(N\cap H\) for any hyperplane. Very strong!
Setting up notation/definitions:
- \(V\) will be a smooth \(n{\hbox{-}}\)manifold
- \(W\) an \(n{\hbox{-}}\)dimensional cobordism
- \(\phi: V\to {\mathbf{R}}\) a smooth function
- \(p\) a critical point of \(\phi\) (i.e. the derivative \(d_p \phi\) vanishes)
- \(H_p\phi = ({{\partial}^2 \phi \over {\partial}x_i^2 {\partial}x_j^2})\) the Hessian matrix
- \(\null_\phi(p)\) the nullity of \(\phi\) at \(p\) is \(\dim \ker H_p\), regarding \(H_p\phi\) as a symmetric bilinear form on \(T_p V\)
- \(p\) is nondegenerate iff \(\null_\phi(p) = 0\).
- The Morse index at \(p\) is the dimension of the maximal subspace on which the associated quadratic form \(H_p \phi\) is negative definite.
- Theorem (Morse Lemma)
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Near a nondegenerate critical point \(p\) of \(\phi\) of index \(k\) there exists a smooth coordinate chart \(U\) and coordinates \(\mathbf{x} \in {\mathbf{R}}^n\) such that \(\phi\) has the form \begin{align*}\phi(\mathbf{x}) = \phi(p) + \mathbf{x}^t A \mathbf{x}\end{align*} where \(A = \operatorname{diag}(-1, \cdots, -1, 1,\cdots 1)\).
Toward a generalization, we can also write \({\mathbf{R}}= {\mathbf{R}}^{k} \times{\mathbf{R}}^{n-k}\) and \begin{align*} \phi(\mathbf{x}_1, \mathbf{x}_2) = \phi(p) - {\left\lVert {\mathbf{x}_1} \right\rVert}^2 + {\left\lVert {\mathbf{x}_2} \right\rVert}^2 \end{align*}
- Lemma (The nondegenerate directions can be split off)
- If \(\null_\phi(p) = \ell\) then we can instead write \({\mathbf{R}}= {\mathbf{R}}^{n-k-\ell} \times{\mathbf{R}}^k \times{\mathbf{R}}^\ell\) and \begin{align*} \phi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + \psi(\mathbf{z}) \end{align*} where \(\psi: {\mathbf{R}}^\ell \to {\mathbf{R}}\) is some smooth function.
- Definition
- A degenerate critical point is embryonic iff \(\null_\phi(p) = 1\) and writing \(L = \ker H_p\phi = \mathop{\mathrm{span}}_{\mathbf{R}}{\mathbf{v}}\), the third directional derivative \(D^3_{\mathbf{v}}\phi\) (?) is nonzero.
We now consider homotopies of Morse functions \(\phi: I \times V \to {\mathbf{R}}\), where we can partially apply the \(I\) factor to get a 1-parameter family \(\left\{{\phi_t {~\mathrel{\Big\vert}~}t\in I}\right\}\).
- Definition
- A homotopy \(\Phi: V\times I \to {\mathbf{R}}\) of Morse functions has a birth-death type critical point at \(p\) at \(t=t_0\) iff \(p\) is embryonic for \(\phi_0\) and \((t_0, p)\) is a nondegenerate critical point of \(\Phi\).
Recall what a Cerf diagram/profile is – I don’t
- Theorem (Whitney)
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In three parts:
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Near an embryonic critical point \(p\) of \(\phi\) of index \(k\) there exist coordinate \((\mathbf{x}, \mathbf{y}, z) \in {\mathbf{R}}^{n-k-1} \oplus {\mathbf{R}}^{k} \oplus {\mathbf{R}}\) such that \(\phi\) has the form \begin{align*} \phi(\mathbf{x}, \mathbf{y}, z) = \phi(p) + {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + z^3 \end{align*}
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If \(p\) is birth-death type for \(\Phi\) of index \(k\), then up to conjugating \(\phi_t\) by a (uniform in \(t\)) family of diffeomorphisms, each \(\phi_t\) is of the form \begin{align*} \phi(\mathbf{x}, \mathbf{y}, z) = \phi(p) + {\left\lVert {\mathbf{x}} \right\rVert}^2 - {\left\lVert {\mathbf{y}} \right\rVert}^2 + z^3 \pm tz \end{align*}
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Any two homotopies \(\Phi, \Phi'\) with points \((p, 0)\) and \((p', 0)\) with the same index and Cerf profile differ only by a diffeomorphism, i.e. there is a family of diffeomorphisms \(h_t\) such that \(\phi'_t \circ h_t = \phi_t\) for every \(t\).
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A generic \(\Phi\) has only nondegenerate and birth-death type singularities.
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- Definition
- A singularity is birth type if the sign on \(t\) is positive, and death type if negative.
- Fact
- Embryonic critical points are isolated, near a birth-type singularity two nondegenerate critical points of indices \(k, k-1\) emerge, and near a death type they merge and disappear.
Pretty vague – I know there are pictures here that make this more obvious, but I couldn’t find much.
- Definition
- A cobordism is a triple \((W; M_+, M_-)\) where \(W\) is an oriented compact smooth manifold with cooriented boundary \({{\partial}}W = M_+ {\textstyle\coprod}M_- = {{\partial}}_- W {\textstyle\coprod}{{\partial}}_+ W\), where the coorientation is provided by the inward (resp. outward) normal vector field (???). We’ll usually just denote this as \(W\).
- Definition
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A Lyapunov cobordism is a triple \((W, \phi , X)\) where
- \(W\) is a usual cobordism,
- \(\phi: W\to {\mathbf{R}}\) is a smooth functional that is constant and has no critical points when restricted to \({{\partial}}W\),
- \(X\) is a gradient-like vector field for \(\phi\) which points inward along \({{\partial}}_- W\) and outward along \({{\partial}}_+ W\).
- Definition
- Such a cobordism is elementary iff there exist no \(X{\hbox{-}}\)trajectories between distinct critical points of \(\phi\).
- Theorem (Smale, h-cobordism)
- Let \(W\) be a cobordism of dimension \(W\geq 6\) such that \(W, {{\partial}}_{\pm}W\) are simply connected, and \(H_*(W, {{\partial}}_- W; {\mathbf{Z}}) = 0\). Then \(W\) admits a Morse function without critical points which is constant on \({{\partial}}_\pm W\).
In particular, \(W \cong I\times M\) is diffeomorphic to the trivial product cobordism, and \(M\cong N\) are diffeomorphic.
Proof (Sketch)
Goal: find a handle decomposition with no handles, then integrate along the gradient vector field of a Morse function \(\phi\) to get a diffeomorphism.
- Find a Morse function and induce a handle decomposition
- Rearrange handles so that lower index handles are attached first
- Define a chain complex as free \({\mathbf{Z}}{\hbox{-}}\)module on handles with boundary given in terms of intersection numbers of attaching spheres \(k\) and belt \(k-1\) spheres
- Find \(k{\hbox{-}}\)handles, create a pair of \(k+1, k+2\) handles such that the \(k+1\) handle cancels/fills in the \(k{\hbox{-}}\)handle (not sure why the \(k+2\) is needed here)
- End up with nothing but an \(n{\hbox{-}}\)handle and an \(n-1{\hbox{-}}\)handle – turn “upside down” and repeat process with \(-\phi\) to remove them.
Proof (Sketch)
- Pick \(\phi: W\to {\mathbf{R}}\) Morse such that \({{\partial}}_\pm W\) are regular level sets.
- Make \(\phi\) self-intersecting (uses a transversality argument)
- Partition manifold into regular level sets \(L_k \coloneqq\phi^{-1}(k - \frac 1 2)\) for each \(k\in {\mathbb{N}}\).
- Letting \(\left\{{p_i}\right\}\) be the critical points in \(L_k\) and \(\left\{{q_j}\right\}\) the critical points in \(L_{k-1}\), form the matrix \(A\) of intersection numbers \(S_{p_i}^- \smile S_{q_j}^+\) between the stable sphere of \(p_i\) and the unstable sphere of \(q_j\).
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Goal: since homology can be read off \(SNF(A)\), and we know \(H_* = 0\) here, we try to reduce \(A\) to SNF with geometric operations
- Handle slides: Add row \(j\) to row \(i\) by moving \(p_i\) to \(L_{k+j}, ~j\geq 1\), deform \(X\) to produce a trajectory \(p_j \to p_i\), then “the stable manifold of \(p_i\) slides over the stable manifold of \(p_j\)” (?) replacing \([S_i^-]\) with \([S_i^-] + [S_j^-]\) in homology.
- This makes \(A = [I, 0; 0, 0]\) a block matrix with the identity in the top-left.
- Handle Cancellation: Take two transverse intersection points \(z_+, z_-\) with local intersection indices \(1, -1\), connect via two paths: one in \(S_i^-\), one in \(S_j^+\). This yields a map \(S^1 \hookrightarrow L_k\), use the Whitney trick to fill with an embedded disc \(\Delta\), then push \(S_i^-\) over \(\Delta\) eliminates \(z_\pm\).
- This leaves a collection \(S_i^-, S_i^+\) for \(i=1,\cdots, r\) intersecting in a single point \(z_0\), then (lemma) there are unique trajectories \(q_i \to p_i\) for each \(I\) and thus they can be eliminated.
- Do this in \(L_k\); we now have a Morse function with no critical points except possibly of index \(0, 1, n-1\), or \(n\).
- Use “Smale’s trick”: trades in an index \(k\) critical point for one of index \(k+1\) and one of index \(k+2\), such that \(k, k+1\) cancel. Trade index \(1\) for index \(2, 3\) and cancel index \(3\) as before.
- Eliminate \(0, n\) with a lemma (unclear)
